2 kinds of integral

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Main Question or Discussion Point

Hi all,

I need help.

What is the difference between

[tex]
\int_{0}^b x^2 dx
[/tex]

with

[tex]
lim_{b\rightarrow \ \infty} \int_{0}^b x^2 dx
[/tex]

???

Can someone please show me algebraically for its clarity?

I don’t understand of the 2nd integral equation means.
Especially the appears of limit as b approach to infinity.

Thanks in advance
 

Answers and Replies

one is a proper integral and the other one is an improper integral
 
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The first integral is a regular definite integral. It basically just measures the area that the parabola [tex]y=x^2[/tex] encloses with the x-axis from the point x=0 to another point x=b.


The second one is an improper integral. THati is:

[tex]\int_{0}^{\infty}x^2dx=\lim_{b\to\infty}\int_{0}^{b}x^2dx[/tex]

SO it means that the function x^2 is unbounde from the above. And if that integral converges, than it means that we are calculating the area that the function [tex]f(x)=x^2[/tex] encloses with the x-axis from x=0 to infinity. In other words the area that the right wing of the parabola encloses with the x-axis.
 

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